Bosonic BdG Hamiltonian

• Bosonic BdG Hamiltonian is a quadratic framework for bosonic systems that incorporates pairing terms and pseudo-Hermitian structure via paraunitary transformations.
• It plays a key role in describing collective excitations, topological amplifiers, and dynamical instabilities across ultracold gases, superconductors, and spin systems.
• Understanding the BdG framework enables practical insights into edge modes, non-Hermitian topology, and advanced numerical techniques for large-scale eigenvalue problems.

A Bosonic Bogoliubov–de Gennes (BdG) Hamiltonian is a general quadratic Hamiltonian describing bosonic quasiparticle excitations in systems with particle non-conserving (pairing) processes. Its central role is in the mean-field and linear-response descriptions of collective modes, superfluidity, and symmetry-breaking phenomena in bosonic systems including ultracold gases, condensed matter, superconductivity, spin/magnonics, and quantum field theory. Unlike its fermionic counterpart, the bosonic BdG Hamiltonian is pseudo-Hermitian (not Hermitian with respect to the usual positive-definite inner product), must be diagonalized using paraunitary transformations, and generically features a spectrum and symmetry structure richer than the standard unitary case. The bosonic BdG framework is also foundational for understanding topology, dynamical instabilities, and emergent phenomena in a wide range of physical contexts.

1. Algebraic Structure and Diagonalization

The bosonic BdG Hamiltonian for a system with one-body Hilbert space $\mathfrak{h}$ is constructed as a quadratic form in creation and annihilation operators: $$H = d\Gamma(h) + \tfrac{1}{2} \left[a^*(k) a^* + a(k^*) a \right]$$ where $h$ is self-adjoint (kinetic/energy term), $k$ is the (possibly unbounded) symmetric pairing operator, and $a^*$, $a$ denote bosonic creation and annihilation operators. Equivalently, in the particle–hole (Nambu) basis, this Hamiltonian is written as

$$\mathcal{H} = \begin{pmatrix} h & k^* \ k & h^T \end{pmatrix}$$

The eigenvalue problem associated with excitations involves the "pseudo-Hermitian" matrix

$$\Sigma_z \mathcal{H} \vec{\Psi} = E \vec{\Psi}, \quad \Sigma_z = \mathrm{diag}(1, -1)$$

This operator is not Hermitian in the standard sense but obeys pseudo-Hermiticity: $$\Sigma_z \mathcal{H}^\dagger \Sigma_z = \mathcal{H}$$ Diagonalization is achieved using a Bogoliubov (paraunitary) transformation $T$ satisfying

$T \Sigma_z T^\dagger = \Sigma_z$

which preserves the canonical commutation relations. The spectrum generically consists of symmetric pairs $E, -E$ and can include non-real (dynamically unstable) eigenvalues.

A pivotal result (Nam et al., 2015, Matsuzawa et al., 2019) is that this diagonalization is well-defined (and implementable on Fock space) if the operator $G = h^{-1/2} J^* k h^{-1/2}$ (with $J$ conjugation) satisfies $\|G\| < 1$ and, for Fock-space implementation, $k h^{-1/2}$ is Hilbert–Schmidt.

2. Pseudo-Hermitian, Krein Space, and Indefinite Metric Structure

The bosonic BdG Hamiltonian acts on a Krein space endowed with an indefinite inner product associated with the metric $\Sigma_z$. In this structure, the eigenvalues' "Krein signature" controls dynamical stability: a dynamically stable Hamiltonian requires that all eigenvalues be real with definite signature (no "Krein collisions"). For finite or infinite systems, the "Krein–unitary" generalization of the Schrieffer–Wolff transformation systematically generates effective Hamiltonians within chosen (signature-definite) subspaces, and preserves stability up to small perturbations (Massarelli et al., 2022).

Pseudo-Hermiticity aligns the bosonic BdG Hamiltonian naturally with non-Hermitian symmetry classes—including the Bernard–LeClair (BL) classification (Lieu, 2018)—permitting the emergence of phenomena (such as exceptional points) absent in standard Hermitian models.

3. Dynamical Instabilities and Topological Amplification

Bosonic BdG systems can exhibit inherent dynamical instability: when a pairing (number non-conserving) perturbation lifts a degeneracy between particle and hole states, the resulting energy splitting can become purely imaginary, leading to modes that grow or decay exponentially in time. This is a direct result of the pseudo-Hermitian structure and the associated indefinite inner product.

Harnessing this instability leads to the concept of a topological amplifier: systems where the bulk spectrum remains stable (real) while edge (boundary) states become dynamically unstable (complex eigenvalues with positive imaginary part), thus allowing for selective parametric amplification localized at the edges (Ling et al., 2020, Ling et al., 2021). The theorem formulated in (Ling et al., 2020) provides the practical selection rule: bulk states with large $|E_0|$ remain stable to first and higher order in the pairing if the "unconventional commutator"

$$\left\lceil A(k), B(k) \right\rfloor = A(k) B(k) - B(k) A^*(-k)$$

vanishes, independent of inversion symmetry or other details.

4. Non-Hermitian Topology: Invariants and Edge Modes

Bosonic BdG Hamiltonians, despite arising from underlying Hermitian physics, possess nontrivial non-Hermitian topology:

• Chern and Winding Numbers: Even in disordered or non-periodic systems, spectral Riesz projections enable the definition of non-Hermitian Chern numbers that classify bulk bands and predict the existence of robust edge states (Peano et al., 2017, Kondo et al., 2020). These invariants are constructed using nonorthogonal projectors and noncommutative geometry:

$$\operatorname{Ch}(Q) = 2\pi i \,\mathcal{T}( Q [V_1 Q, V_2 Q] )$$

for a projection $Q$ and trace per unit volume $\mathcal{T}$.

• Berry Connection and $\mathbb{Z}_2$ Indices: For systems with pseudo-time-reversal symmetry (ensured by, e.g., antiferromagnetic order), a bosonic version of Kramers pairs leads to $\mathbb{Z}_2$ topological invariants, constructed from the paraunitary eigenvectors and the "bosonic Berry connection"

$$\mathcal{A}_{n,\sigma}(\mathbf{k}) = i \sigma v^*_{n,\sigma}(\mathbf{k}) \Sigma_z \nabla_\mathbf{k} v_{n,\sigma}(\mathbf{k})$$

with edge or surface states physically linked to nontrivial $D_n = 1$ (Kondo et al., 2020).

• Boundary-Dynamical Instability and Skin Effect: Non-Bloch band theory (Yokomizo et al., 2020) clarifies that, given the non-Hermiticity of $\tau_z H_\mathrm{BdG}$, bulk spectra are strongly boundary-condition dependent. The non-Hermitian skin effect—where open boundary conditions lead to exponential localization of bulk eigenstates—emerges generically, and the spectrum must be computed using a generalized Brillouin zone (GBZ) defined via the magnitude of roots of the characteristic equation:

$|\beta_{2N}| = |\beta_{2N+1}|$

Results demonstrate that the skin effect can be modulated (even eliminated) by tuning parameters such as the chemical potential, resulting in reentrant behavior of localization.

• Defective Flat Bands and Emergent $\mathbb{Z}_2$ Topology: Certain BdG models realize a flat-band limit in which the BdG matrix becomes defective. Here, the Berry phase is $\mathbb{Z}_2$ quantized due to emergent PT symmetry, even when conventional bulk invariants are trivial. These features underlie bosonic analogs of Andreev bound states, with boundary states pinned outside the bulk bands and encoded by nontrivial Berry phase (polarization) of the hole sector (Okuma, 2023).

5. Applications: Excitations, Superfluidity, and Many-Body Physics

The bosonic BdG formalism is central to describing elementary and collective excitations (elementary modes, collective oscillations, vortices) in various correlated systems:

• In Bose–Einstein condensates, BdG theory linearizes the time-dependent Gross–Pitaevskii (GP) or Hartree–Fock–Bogoliubov (HFB) equations about the ground state, yielding a self-consistent quadratic (quasifree) Hamiltonian whose spectrum defines the excitation energies (e.g., Goldstone and amplitude modes) (Bach et al., 2016).
• The LPDA equation, derived via double coarse graining of the full BdG gap equation (Simonucci et al., 2014), provides a nonlinear, local, gauge-invariant differential equation for the gap parameter $\Delta(\mathbf{r})$ that interpolates between the Ginzburg–Landau and Gross–Pitaevskii equations. This equation is computationally far less demanding than full BdG diagonalization yet retains quantitative accuracy across the BCS–BEC crossover (excluding non-smooth gap profiles at deep BCS).
• Exact diagonalization of truncated BdG Hamiltonians in interacting boson models (Ferrari, 2016, Ferrari, 2016) reveals the detailed structure of excitation spectra, including distinctions between symmetric "s-pseudobosons" (zero-momentum collective excitations) and $\eta$-pseudobosons (finite-momentum/dissipative modes), the latter of which explain observed dissipation mechanisms not captured by mean-field or BCA/GPT approximations.
• The explicit algebraic construction for diagonalizing BdG Hamiltonians—including the implementation of Bogoliubov transformations, extraction of ground state energy shifts, and the formulation of mean-field potentials in coupled electron–phonon and interacting Bose–Fermi models—allows a general quadratic Hamiltonian to be mapped to a direct sum of normal modes plus a renormalized vacuum shift (Wang et al., 2018, Matsuzawa et al., 2019).

6. Numerical and Computational Techniques

Modern computational treatment of bosonic BdG Hamiltonians leverages analytical and numerical methods specialized for large-scale, indefinite, non-Hermitian eigenvalue problems:

• Fourier Spectral Algorithms: For spatially extended bosonic systems (e.g., spin-1 BECs in harmonic traps), a matrix-free Fourier spectral algorithm combines pseudospectral discretization and FFT-accelerated operator evaluation with Gram–Schmidt bi-orthogonalization for subspace iteration. This yields efficient, numerically stable, and spectrally convergent solvers for BdG eigenproblems in 1–3D (Li et al., 10 Jun 2025). Spectral convergence is guaranteed under smoothness conditions (with controlled error estimates), and large-scale problems can be solved with modest memory resources.
• Phase-Space Construction for Multiband and Lattice Systems: In multi-band or spatially complex lattices with Wannier obstructions (e.g., square-octagon lattices), the phase-space construction embeds real-space Wannier orbital and local symmetry data into an extended Bloch basis at each momentum. This approach clarifies the interplay between local orbital irreducible representations, global superconducting coherence, and pairing symmetry, enabling systematic treatment of unconventional and topological superconducting states (Sharma et al., 28 Dec 2024).

7. Summary Table of Key Mathematical Structures in Bosonic BdG Hamiltonians

Structure	Description / Role	BdG-specific Features
Quadratic Hamiltonian	$H = d\Gamma(h) + \text{pairing}$	Contains anomalous (pairing) terms
Krein (Indefinite) Metric	$\Sigma_z$ or $\tau_z$; $$\Sigma_z \mathcal{H}^\dagger \Sigma_z = \mathcal{H}$$	Pseudo-Hermiticity; stability via signature
Paraunitary Transformation	$T \Sigma_z T^\dagger = \Sigma_z$	Preserves bosonic commutators
Topological Invariants	Chern number, $\mathbb{Z}_2$, winding number	Defined via non-Hermitian projectors
Non-Hermitian Skin Effect	Boundary-localized bulk spectrum (open b.c.)	Arises due to non-Hermiticity of $\Sigma_z \mathcal{H}$
Dynamical Instability	Imaginary eigenvalues lead to amplification/decay	Controlled via symmetry, commutators

References to Key Results

• (Simonucci et al., 2014): LPDA (local phase density approximation) equation for superfluid gap, BCS–BEC crossover, and computational efficiency.
• (Nam et al., 2015, Matsuzawa et al., 2019): Rigorous diagonalization and ground state energy of quadratic bosonic Hamiltonians.
• (Bach et al., 2016): Time-dependent Hartree–Fock–Bogoliubov equations and connection to BdG linearization.
• (Ferrari, 2016, Ferrari, 2016): Exact solution and spectrum structure for truncated BdG Hamiltonians (pseudobosons).
• (Peano et al., 2017, Lieu, 2018, Kondo et al., 2020, Yokomizo et al., 2020, Okuma, 2022): Non-Hermitian topology, skin effect, and edge mode instabilities; paraunitary Berry connection and BRZ-GBZ framework.
• (Ling et al., 2020, Ling et al., 2021): Symmetry selection rules for stability, topological amplifiers, unconventional commutator.
• (Massarelli et al., 2022): Krein-unitary Schrieffer-Wolff perturbation theory for BdG band touchings.
• (Okuma, 2023): Defective flat-band limit, emergent $\mathbb{Z}_2$ quantization in topological bound states.
• (Sharma et al., 28 Dec 2024): Phase-space approach for BdG, Wannier pairing, and complex lattice symmetry.
• (Li et al., 10 Jun 2025): Efficient Fourier spectral methods for large-scale bosonic BdG eigenvalue problems.

These mathematical and physical structures collectively define the bosonic BdG Hamiltonian as an indispensable tool for the paper of collective phenomena in bosonic many-body quantum systems and as a fertile framework for exploring non-Hermitian topology and edge dynamics.